Summary: Formulas are obtained for the simultaneous determination of two or three unknown thermal coefficients of a semi-infinite material with temperature-dependent thermal conductivity through a phase-change process with an over-specified condition on the fixed face through a moving boundary problem (inverse Stefan problem) or a free boundary problem (Stefan problem), respectively. We complete and improve the analysis done in [the second author, “The determination of unknown thermal coefficients through phase change process with temperature-dependent thermal conductivity", Int. Commun. Heat Mass Transfer 25, No.~1, 139‒147 (1998), \url{doi:10.1016/S0735-1933(97)00145-0}] and we also study the sensitivity of the solution depending on different thermal parameters, applied to aluminium.